- FindStat

Your data matches 9 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001250

Mapping

Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001250: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => 3
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => 3
[1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => 3
[1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => 2
[1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => 2
[1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => 2
[1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => 2
[1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => 4
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1]
 => 4
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1]
 => 4
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1]
 => 4
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1]
 => 3
[1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1]
 => 3
[1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1]
 => 3
[1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [2,2,2]
 => 3
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [2,2,2,1,1]
 => 5
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1]
 => 4
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1]
 => 4
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1]
 => 4
[1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1]
 => 3
[1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1]
 => 3
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1]
 => 3
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [2,2,2]
 => 3
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [2,2,2,1,1]
 => 5
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1]
 => 4
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1]
 => 4
[1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1]
 => 3
[1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1]
 => 3
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1]
 => 3
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [2,2,2]
 => 3
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [2,2,2,1,1]
 => 5
[1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1]
 => 2
[1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1]
 => 2
[1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1]
 => 2
[1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1]
 => 2
[1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1]
 => 2
[1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1]
 => 2
[1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1]
 => 2
[1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1]
 => 2
[1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [2,2,1]
 => 3
[1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [2,2]
 => 2
[1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [2,2]
 => 2
[1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [2,2]
 => 2
[1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [3,3]
 => 0
[1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [3,3,1,1,1]
 => 3
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [2,2,1,1,1]
 => 5
[1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [2,2,1,1,1]
 => 5
[1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [2,2,1,1,1]
 => 5
[1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [3,3,1,1]
 => 2

Description

The number of parts of a partition that are not congruent 0 modulo 3.

Matching statistic: St000329

Mapping

Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000329: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 20%

Values

[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 3
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => ? = 3
[1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => ? = 3
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2
[1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 2
[1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 2
[1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 2
[1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 4
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [3,3,3,2,1]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 4
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [5,2,2,2,1]
 => [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 4
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
 => ? = 4
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 3
[1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => [5,2,2,1]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => ? = 3
[1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => [5,4,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 3
[1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [5,4,3]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 3
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2,1]
 => [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 5
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [4,3,3,2,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 4
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,4,2,2,1]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
 => ? = 4
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [4,4,3,1,1]
 => [1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
 => ? = 4
[1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [4,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => ? = 3
[1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [5,3,1,1]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => ? = 3
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => [5,4,2]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => ? = 3
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2]
 => [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
 => ? = 3
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2,1]
 => [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 5
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,2,1]
 => [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
 => ? = 4
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1,1]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => ? = 4
[1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [4,3,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => ? = 3
[1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => [5,3,2]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => ? = 3
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => [4,4,3,1]
 => [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
 => ? = 3
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2]
 => [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => ? = 3
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2,1]
 => [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
 => ? = 5
[1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2
[1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
 => ? = 2
[1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 2
[1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => [4,3,2,2]
 => [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => ? = 2
[1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,1]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
 => ? = 2
[1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 2
[1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [4,3,3,1]
 => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => ? = 2
[1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [5,3,2,2]
 => [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => ? = 2
[1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,1]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
 => ? = 3
[1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 2
[1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => [4,4,2,1]
 => [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
 => ? = 2
[1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => ? = 2
[1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2]
 => [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => ? = 0
[1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,1]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
 => ? = 3
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 5
[1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 5
[1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
 => ? = 5
[1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
 => ? = 2
[1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2]
 => [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
 => ? = 2
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1]
 => [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 3
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [3,3,3,3,2,1]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
 => ? = 5
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [6,2,2,2,2,1]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,0]
 => ? = 5
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [6,5,1,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
 => ? = 5
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [3,3,3,2,1]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 4
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
 => [6,2,2,2,1]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 4
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
 => [6,5,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
 => ? = 4
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [6,5,4]
 => [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => ? = 4
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [4,4,4,3,2,1]
 => [1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,1,0,0]
 => ? = 6
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [4,4,3,3,2,1]
 => [1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
 => ? = 5
[1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 3
[1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2
[1,1,1,1,0,0,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 2
[1,1,1,1,0,0,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 2
[1,1,1,1,0,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 2
[1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 3
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2
[1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 2
[1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 2
[1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
 => [4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 2
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2
[1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0,0]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 2

Description

The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1.

Matching statistic: St001192

Mapping

Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001192: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 20%

Values

[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 3
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => ? = 3
[1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => ? = 3
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2
[1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 2
[1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 2
[1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 2
[1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 4
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [3,3,3,2,1]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 4
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [5,2,2,2,1]
 => [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 4
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
 => ? = 4
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 3
[1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => [5,2,2,1]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => ? = 3
[1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => [5,4,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 3
[1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [5,4,3]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 3
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2,1]
 => [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 5
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [4,3,3,2,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 4
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,4,2,2,1]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
 => ? = 4
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [4,4,3,1,1]
 => [1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
 => ? = 4
[1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [4,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => ? = 3
[1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [5,3,1,1]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => ? = 3
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => [5,4,2]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => ? = 3
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2]
 => [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
 => ? = 3
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2,1]
 => [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 5
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,2,1]
 => [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
 => ? = 4
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1,1]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => ? = 4
[1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [4,3,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => ? = 3
[1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => [5,3,2]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => ? = 3
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => [4,4,3,1]
 => [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
 => ? = 3
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2]
 => [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => ? = 3
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2,1]
 => [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
 => ? = 5
[1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2
[1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
 => ? = 2
[1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 2
[1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => [4,3,2,2]
 => [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => ? = 2
[1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,1]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
 => ? = 2
[1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 2
[1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [4,3,3,1]
 => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => ? = 2
[1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [5,3,2,2]
 => [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => ? = 2
[1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,1]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
 => ? = 3
[1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 2
[1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => [4,4,2,1]
 => [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
 => ? = 2
[1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => ? = 2
[1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2]
 => [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => ? = 0
[1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,1]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
 => ? = 3
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 5
[1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 5
[1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
 => ? = 5
[1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
 => ? = 2
[1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2]
 => [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
 => ? = 2
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1]
 => [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 3
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [3,3,3,3,2,1]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
 => ? = 5
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [6,2,2,2,2,1]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,0]
 => ? = 5
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [6,5,1,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
 => ? = 5
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [3,3,3,2,1]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 4
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
 => [6,2,2,2,1]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 4
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
 => [6,5,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
 => ? = 4
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [6,5,4]
 => [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => ? = 4
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [4,4,4,3,2,1]
 => [1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,1,0,0]
 => ? = 6
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [4,4,3,3,2,1]
 => [1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
 => ? = 5
[1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 3
[1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2
[1,1,1,1,0,0,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 2
[1,1,1,1,0,0,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 2
[1,1,1,1,0,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 2
[1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 3
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2
[1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 2
[1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 2
[1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
 => [4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 2
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2
[1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0,0]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 2

Description

The maximal dimension of

$Ext_A^2(S,A)$ for a simple module

$S$ over the corresponding Nakayama algebra

$A$ .

Matching statistic: St001210

Mapping

Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001210: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 20%

Values

[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 3
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => ? = 3
[1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => ? = 3
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2
[1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 2
[1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 2
[1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 2
[1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 4
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [3,3,3,2,1]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 4
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [5,2,2,2,1]
 => [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 4
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
 => ? = 4
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 3
[1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => [5,2,2,1]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => ? = 3
[1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => [5,4,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 3
[1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [5,4,3]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 3
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2,1]
 => [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 5
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [4,3,3,2,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 4
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,4,2,2,1]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
 => ? = 4
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [4,4,3,1,1]
 => [1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
 => ? = 4
[1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [4,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => ? = 3
[1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [5,3,1,1]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => ? = 3
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => [5,4,2]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => ? = 3
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2]
 => [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
 => ? = 3
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2,1]
 => [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 5
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,2,1]
 => [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
 => ? = 4
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1,1]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => ? = 4
[1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [4,3,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => ? = 3
[1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => [5,3,2]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => ? = 3
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => [4,4,3,1]
 => [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
 => ? = 3
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2]
 => [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => ? = 3
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2,1]
 => [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
 => ? = 5
[1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2
[1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
 => ? = 2
[1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 2
[1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => [4,3,2,2]
 => [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => ? = 2
[1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,1]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
 => ? = 2
[1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 2
[1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [4,3,3,1]
 => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => ? = 2
[1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [5,3,2,2]
 => [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => ? = 2
[1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,1]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
 => ? = 3
[1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 2
[1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => [4,4,2,1]
 => [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
 => ? = 2
[1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => ? = 2
[1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2]
 => [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => ? = 0
[1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,1]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
 => ? = 3
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 5
[1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 5
[1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
 => ? = 5
[1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
 => ? = 2
[1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2]
 => [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
 => ? = 2
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1]
 => [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 3
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [3,3,3,3,2,1]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
 => ? = 5
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [6,2,2,2,2,1]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,0]
 => ? = 5
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [6,5,1,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
 => ? = 5
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [3,3,3,2,1]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 4
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
 => [6,2,2,2,1]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 4
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
 => [6,5,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
 => ? = 4
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [6,5,4]
 => [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => ? = 4
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [4,4,4,3,2,1]
 => [1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,1,0,0]
 => ? = 6
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [4,4,3,3,2,1]
 => [1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
 => ? = 5
[1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 3
[1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2
[1,1,1,1,0,0,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 2
[1,1,1,1,0,0,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 2
[1,1,1,1,0,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 2
[1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 3
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2
[1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 2
[1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 2
[1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
 => [4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 2
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2
[1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0,0]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 2

Description

Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path.

Matching statistic: St001215

Mapping

Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001215: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 20%

Values

[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 4 = 3 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => ? = 3 + 1
[1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => ? = 3 + 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 3 = 2 + 1
[1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 3 = 2 + 1
[1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 3 = 2 + 1
[1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 3 = 2 + 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 4 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [3,3,3,2,1]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 4 + 1
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [5,2,2,2,1]
 => [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 4 + 1
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
 => ? = 4 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 4 = 3 + 1
[1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => [5,2,2,1]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => ? = 3 + 1
[1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => [5,4,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 3 + 1
[1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [5,4,3]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 3 + 1
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2,1]
 => [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 5 + 1
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [4,3,3,2,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 4 + 1
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,4,2,2,1]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
 => ? = 4 + 1
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [4,4,3,1,1]
 => [1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
 => ? = 4 + 1
[1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [4,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => ? = 3 + 1
[1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [5,3,1,1]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => ? = 3 + 1
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => [5,4,2]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => ? = 3 + 1
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2]
 => [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
 => ? = 3 + 1
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2,1]
 => [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 5 + 1
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,2,1]
 => [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
 => ? = 4 + 1
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1,1]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => ? = 4 + 1
[1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [4,3,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => ? = 3 + 1
[1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => [5,3,2]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => ? = 3 + 1
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => [4,4,3,1]
 => [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
 => ? = 3 + 1
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2]
 => [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => ? = 3 + 1
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2,1]
 => [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
 => ? = 5 + 1
[1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 3 = 2 + 1
[1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
 => ? = 2 + 1
[1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 3 = 2 + 1
[1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => [4,3,2,2]
 => [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => ? = 2 + 1
[1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,1]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
 => ? = 2 + 1
[1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 3 = 2 + 1
[1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [4,3,3,1]
 => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => ? = 2 + 1
[1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [5,3,2,2]
 => [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => ? = 2 + 1
[1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,1]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
 => ? = 3 + 1
[1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 3 = 2 + 1
[1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => [4,4,2,1]
 => [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
 => ? = 2 + 1
[1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => ? = 2 + 1
[1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2]
 => [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => ? = 0 + 1
[1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,1]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
 => ? = 3 + 1
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 5 + 1
[1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 5 + 1
[1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
 => ? = 5 + 1
[1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
 => ? = 2 + 1
[1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2]
 => [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
 => ? = 2 + 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1]
 => [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 3 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [3,3,3,3,2,1]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
 => ? = 5 + 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [6,2,2,2,2,1]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,0]
 => ? = 5 + 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [6,5,1,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
 => ? = 5 + 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [3,3,3,2,1]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 4 + 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
 => [6,2,2,2,1]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 4 + 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
 => [6,5,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
 => ? = 4 + 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [6,5,4]
 => [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => ? = 4 + 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [4,4,4,3,2,1]
 => [1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,1,0,0]
 => ? = 6 + 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [4,4,3,3,2,1]
 => [1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
 => ? = 5 + 1
[1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 4 = 3 + 1
[1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 3 = 2 + 1
[1,1,1,1,0,0,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 3 = 2 + 1
[1,1,1,1,0,0,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 3 = 2 + 1
[1,1,1,1,0,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 4 = 3 + 1
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 3 = 2 + 1
[1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 3 = 2 + 1
[1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 3 = 2 + 1
[1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
 => [4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 3 = 2 + 1
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 3 = 2 + 1
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 3 = 2 + 1
[1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0,0]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 3 = 2 + 1

Description

Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. Then the statistic gives the vector space dimension of the second Ext-group between X and the regular module. For the first 196 values, the statistic also gives the number of indecomposable non-projective modules

$X$ such that

$\tau(X)$ has codominant dimension equal to one and projective dimension equal to one.

Matching statistic: St001233

Mapping

Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001233: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 20%

Values

[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => ? = 3 - 1
[1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => ? = 3 - 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 1 = 2 - 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 4 - 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [3,3,3,2,1]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 4 - 1
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [5,2,2,2,1]
 => [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 4 - 1
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
 => ? = 4 - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => [5,2,2,1]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => ? = 3 - 1
[1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => [5,4,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 3 - 1
[1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [5,4,3]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 3 - 1
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2,1]
 => [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 5 - 1
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [4,3,3,2,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 4 - 1
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,4,2,2,1]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
 => ? = 4 - 1
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [4,4,3,1,1]
 => [1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
 => ? = 4 - 1
[1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [4,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => ? = 3 - 1
[1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [5,3,1,1]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => ? = 3 - 1
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => [5,4,2]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => ? = 3 - 1
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2]
 => [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
 => ? = 3 - 1
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2,1]
 => [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 5 - 1
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,2,1]
 => [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
 => ? = 4 - 1
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1,1]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => ? = 4 - 1
[1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [4,3,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => ? = 3 - 1
[1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => [5,3,2]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => ? = 3 - 1
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => [4,4,3,1]
 => [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
 => ? = 3 - 1
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2]
 => [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => ? = 3 - 1
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2,1]
 => [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
 => ? = 5 - 1
[1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
 => ? = 2 - 1
[1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => [4,3,2,2]
 => [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => ? = 2 - 1
[1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,1]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
 => ? = 2 - 1
[1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [4,3,3,1]
 => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => ? = 2 - 1
[1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [5,3,2,2]
 => [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => ? = 2 - 1
[1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,1]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
 => ? = 3 - 1
[1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 1 = 2 - 1
[1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => [4,4,2,1]
 => [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
 => ? = 2 - 1
[1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => ? = 2 - 1
[1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2]
 => [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => ? = 0 - 1
[1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,1]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
 => ? = 3 - 1
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 5 - 1
[1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 5 - 1
[1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
 => ? = 5 - 1
[1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
 => ? = 2 - 1
[1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2]
 => [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
 => ? = 2 - 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1]
 => [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 3 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [3,3,3,3,2,1]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
 => ? = 5 - 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [6,2,2,2,2,1]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,0]
 => ? = 5 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [6,5,1,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
 => ? = 5 - 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [3,3,3,2,1]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 4 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
 => [6,2,2,2,1]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 4 - 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
 => [6,5,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
 => ? = 4 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [6,5,4]
 => [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => ? = 4 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [4,4,4,3,2,1]
 => [1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,1,0,0]
 => ? = 6 - 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [4,4,3,3,2,1]
 => [1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
 => ? = 5 - 1
[1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 2 = 3 - 1
[1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,1,1,1,0,0,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,1,1,1,0,0,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,1,1,1,0,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 2 = 3 - 1
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
 => [4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0,0]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 1 = 2 - 1

Description

The number of indecomposable 2-dimensional modules with projective dimension one.

Matching statistic: St001481

Mapping

Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001481: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 20%

Values

[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => ? = 3 - 1
[1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => ? = 3 - 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 1 = 2 - 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 4 - 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [3,3,3,2,1]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 4 - 1
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [5,2,2,2,1]
 => [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 4 - 1
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
 => ? = 4 - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => [5,2,2,1]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => ? = 3 - 1
[1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => [5,4,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 3 - 1
[1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [5,4,3]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 3 - 1
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2,1]
 => [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 5 - 1
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [4,3,3,2,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 4 - 1
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,4,2,2,1]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
 => ? = 4 - 1
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [4,4,3,1,1]
 => [1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
 => ? = 4 - 1
[1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [4,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => ? = 3 - 1
[1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [5,3,1,1]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => ? = 3 - 1
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => [5,4,2]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => ? = 3 - 1
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2]
 => [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
 => ? = 3 - 1
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2,1]
 => [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 5 - 1
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,2,1]
 => [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
 => ? = 4 - 1
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1,1]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => ? = 4 - 1
[1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [4,3,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => ? = 3 - 1
[1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => [5,3,2]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => ? = 3 - 1
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => [4,4,3,1]
 => [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
 => ? = 3 - 1
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2]
 => [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => ? = 3 - 1
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2,1]
 => [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
 => ? = 5 - 1
[1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
 => ? = 2 - 1
[1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => [4,3,2,2]
 => [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => ? = 2 - 1
[1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,1]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
 => ? = 2 - 1
[1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [4,3,3,1]
 => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => ? = 2 - 1
[1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [5,3,2,2]
 => [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => ? = 2 - 1
[1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,1]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
 => ? = 3 - 1
[1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 1 = 2 - 1
[1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => [4,4,2,1]
 => [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
 => ? = 2 - 1
[1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => ? = 2 - 1
[1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2]
 => [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => ? = 0 - 1
[1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,1]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
 => ? = 3 - 1
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 5 - 1
[1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 5 - 1
[1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
 => ? = 5 - 1
[1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
 => ? = 2 - 1
[1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2]
 => [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
 => ? = 2 - 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1]
 => [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 3 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [3,3,3,3,2,1]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
 => ? = 5 - 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [6,2,2,2,2,1]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,0]
 => ? = 5 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [6,5,1,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
 => ? = 5 - 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [3,3,3,2,1]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 4 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
 => [6,2,2,2,1]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 4 - 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
 => [6,5,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
 => ? = 4 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [6,5,4]
 => [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => ? = 4 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [4,4,4,3,2,1]
 => [1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,1,0,0]
 => ? = 6 - 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [4,4,3,3,2,1]
 => [1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
 => ? = 5 - 1
[1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 2 = 3 - 1
[1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,1,1,1,0,0,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,1,1,1,0,0,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,1,1,1,0,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 2 = 3 - 1
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
 => [4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0,0]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 1 = 2 - 1

Description

The minimal height of a peak of a Dyck path.

Matching statistic: St001231

Mapping

Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001231: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 20%

Values

[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 0 = 2 - 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 1 = 3 - 2
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => ? = 3 - 2
[1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => ? = 3 - 2
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 0 = 2 - 2
[1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 0 = 2 - 2
[1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 0 = 2 - 2
[1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 0 = 2 - 2
[1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 4 - 2
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [3,3,3,2,1]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 4 - 2
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [5,2,2,2,1]
 => [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 4 - 2
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
 => ? = 4 - 2
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 1 = 3 - 2
[1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => [5,2,2,1]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => ? = 3 - 2
[1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => [5,4,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 3 - 2
[1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [5,4,3]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 3 - 2
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2,1]
 => [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 5 - 2
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [4,3,3,2,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 4 - 2
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,4,2,2,1]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
 => ? = 4 - 2
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [4,4,3,1,1]
 => [1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
 => ? = 4 - 2
[1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [4,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => ? = 3 - 2
[1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [5,3,1,1]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => ? = 3 - 2
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => [5,4,2]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => ? = 3 - 2
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2]
 => [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
 => ? = 3 - 2
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2,1]
 => [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 5 - 2
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,2,1]
 => [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
 => ? = 4 - 2
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1,1]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => ? = 4 - 2
[1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [4,3,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => ? = 3 - 2
[1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => [5,3,2]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => ? = 3 - 2
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => [4,4,3,1]
 => [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
 => ? = 3 - 2
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2]
 => [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => ? = 3 - 2
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2,1]
 => [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
 => ? = 5 - 2
[1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 0 = 2 - 2
[1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
 => ? = 2 - 2
[1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 0 = 2 - 2
[1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => [4,3,2,2]
 => [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => ? = 2 - 2
[1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,1]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
 => ? = 2 - 2
[1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 0 = 2 - 2
[1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [4,3,3,1]
 => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => ? = 2 - 2
[1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [5,3,2,2]
 => [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => ? = 2 - 2
[1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,1]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
 => ? = 3 - 2
[1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 0 = 2 - 2
[1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => [4,4,2,1]
 => [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
 => ? = 2 - 2
[1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => ? = 2 - 2
[1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2]
 => [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => ? = 0 - 2
[1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,1]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
 => ? = 3 - 2
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 5 - 2
[1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 5 - 2
[1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
 => ? = 5 - 2
[1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
 => ? = 2 - 2
[1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2]
 => [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
 => ? = 2 - 2
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1]
 => [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 3 - 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [3,3,3,3,2,1]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
 => ? = 5 - 2
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [6,2,2,2,2,1]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,0]
 => ? = 5 - 2
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [6,5,1,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
 => ? = 5 - 2
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [3,3,3,2,1]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 4 - 2
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
 => [6,2,2,2,1]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 4 - 2
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
 => [6,5,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
 => ? = 4 - 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [6,5,4]
 => [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => ? = 4 - 2
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [4,4,4,3,2,1]
 => [1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,1,0,0]
 => ? = 6 - 2
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [4,4,3,3,2,1]
 => [1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
 => ? = 5 - 2
[1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 1 = 3 - 2
[1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 0 = 2 - 2
[1,1,1,1,0,0,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 0 = 2 - 2
[1,1,1,1,0,0,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 0 = 2 - 2
[1,1,1,1,0,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 0 = 2 - 2
[1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 1 = 3 - 2
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 0 = 2 - 2
[1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 0 = 2 - 2
[1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 0 = 2 - 2
[1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
 => [4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 0 = 2 - 2
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 0 = 2 - 2
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 0 = 2 - 2
[1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0,0]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 0 = 2 - 2

Description

The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. Actually the same statistics results for algebras with at most 7 simple modules when dropping the assumption that the module has projective dimension one. The author is not sure whether this holds in general.

Matching statistic: St001234

Mapping

Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001234: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 20%

Values

[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 0 = 2 - 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 1 = 3 - 2
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => ? = 3 - 2
[1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => ? = 3 - 2
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 0 = 2 - 2
[1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 0 = 2 - 2
[1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 0 = 2 - 2
[1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 0 = 2 - 2
[1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 4 - 2
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [3,3,3,2,1]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 4 - 2
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [5,2,2,2,1]
 => [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 4 - 2
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
 => ? = 4 - 2
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 1 = 3 - 2
[1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => [5,2,2,1]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => ? = 3 - 2
[1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => [5,4,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 3 - 2
[1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [5,4,3]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 3 - 2
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2,1]
 => [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 5 - 2
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [4,3,3,2,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 4 - 2
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,4,2,2,1]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
 => ? = 4 - 2
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [4,4,3,1,1]
 => [1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
 => ? = 4 - 2
[1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [4,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => ? = 3 - 2
[1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [5,3,1,1]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => ? = 3 - 2
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => [5,4,2]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => ? = 3 - 2
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2]
 => [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
 => ? = 3 - 2
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2,1]
 => [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 5 - 2
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,2,1]
 => [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
 => ? = 4 - 2
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1,1]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => ? = 4 - 2
[1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [4,3,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => ? = 3 - 2
[1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => [5,3,2]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => ? = 3 - 2
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => [4,4,3,1]
 => [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
 => ? = 3 - 2
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2]
 => [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => ? = 3 - 2
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2,1]
 => [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
 => ? = 5 - 2
[1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 0 = 2 - 2
[1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
 => ? = 2 - 2
[1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 0 = 2 - 2
[1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => [4,3,2,2]
 => [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => ? = 2 - 2
[1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,1]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
 => ? = 2 - 2
[1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 0 = 2 - 2
[1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [4,3,3,1]
 => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => ? = 2 - 2
[1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [5,3,2,2]
 => [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => ? = 2 - 2
[1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,1]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
 => ? = 3 - 2
[1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 0 = 2 - 2
[1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => [4,4,2,1]
 => [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
 => ? = 2 - 2
[1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => ? = 2 - 2
[1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2]
 => [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => ? = 0 - 2
[1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,1]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
 => ? = 3 - 2
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 5 - 2
[1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 5 - 2
[1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
 => ? = 5 - 2
[1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
 => ? = 2 - 2
[1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2]
 => [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
 => ? = 2 - 2
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1]
 => [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 3 - 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [3,3,3,3,2,1]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
 => ? = 5 - 2
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [6,2,2,2,2,1]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,0]
 => ? = 5 - 2
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [6,5,1,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
 => ? = 5 - 2
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [3,3,3,2,1]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 4 - 2
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
 => [6,2,2,2,1]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 4 - 2
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
 => [6,5,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
 => ? = 4 - 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [6,5,4]
 => [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => ? = 4 - 2
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [4,4,4,3,2,1]
 => [1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,1,0,0]
 => ? = 6 - 2
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [4,4,3,3,2,1]
 => [1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
 => ? = 5 - 2
[1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 1 = 3 - 2
[1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 0 = 2 - 2
[1,1,1,1,0,0,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 0 = 2 - 2
[1,1,1,1,0,0,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 0 = 2 - 2
[1,1,1,1,0,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 0 = 2 - 2
[1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 1 = 3 - 2
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 0 = 2 - 2
[1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 0 = 2 - 2
[1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 0 = 2 - 2
[1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
 => [4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 0 = 2 - 2
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 0 = 2 - 2
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 0 = 2 - 2
[1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0,0]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 0 = 2 - 2

Description

The number of indecomposable three dimensional modules with projective dimension one. It return zero when there are no such modules.